Properties

Label 2-966-23.3-c1-0-11
Degree $2$
Conductor $966$
Sign $0.805 + 0.593i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.142 − 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (2.81 − 3.25i)5-s + (0.959 + 0.281i)6-s + (−0.841 + 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−3.62 − 2.32i)10-s + (−0.684 + 4.76i)11-s + (0.142 − 0.989i)12-s + (4.97 + 3.19i)13-s + (0.654 + 0.755i)14-s + (1.78 + 3.91i)15-s + (0.841 − 0.540i)16-s + (1.85 + 0.543i)17-s + ⋯
L(s)  = 1  + (−0.100 − 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (1.26 − 1.45i)5-s + (0.391 + 0.115i)6-s + (−0.317 + 0.204i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (−1.14 − 0.735i)10-s + (−0.206 + 1.43i)11-s + (0.0410 − 0.285i)12-s + (1.37 + 0.886i)13-s + (0.175 + 0.201i)14-s + (0.461 + 1.01i)15-s + (0.210 − 0.135i)16-s + (0.448 + 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.805 + 0.593i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.805 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62024 - 0.532367i\)
\(L(\frac12)\) \(\approx\) \(1.62024 - 0.532367i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 + 0.989i)T \)
3 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (3.23 - 3.53i)T \)
good5 \( 1 + (-2.81 + 3.25i)T + (-0.711 - 4.94i)T^{2} \)
11 \( 1 + (0.684 - 4.76i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-4.97 - 3.19i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-1.85 - 0.543i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-6.37 + 1.87i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (2.53 + 0.744i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-1.79 - 3.93i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (0.811 + 0.936i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-0.677 + 0.781i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-4.72 + 10.3i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 9.77T + 47T^{2} \)
53 \( 1 + (7.78 - 5.00i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (0.705 + 0.453i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (2.79 + 6.10i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.587 - 4.08i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (1.03 + 7.20i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-0.579 + 0.170i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-9.86 - 6.34i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (9.84 + 11.3i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-3.74 + 8.19i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-4.26 + 4.92i)T + (-13.8 - 96.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.848754552422843329177943939536, −9.187268574527249628988602565180, −8.896586652767183688724868850509, −7.56161609272633567949928603967, −6.15787244871223323032015290714, −5.40259762657539428269825163867, −4.67744876070587464576334039483, −3.70990974704593160328404084291, −2.12251114766481148834303213851, −1.21410053541546668654100033144, 1.10381848466465554984276813760, 2.81355988801719464133762085464, 3.53334160231521092659118719197, 5.59841742298348369442071847948, 5.90920258679881357890457210467, 6.47148005792035399188638849630, 7.50692639842723479830549969465, 8.175347941683878416385131680964, 9.325038512534127974094020629938, 10.12498383980657533637089094101

Graph of the $Z$-function along the critical line